Mathematics, 23.04.2020 01:05 Mickey2018

A random sample of 110 lightning flashes in a certain region resulted in a sample average radar echo duration of 0.81 second and a sample standard deviation of 0.34 second. (a) Calculate a 99% confidence interval for the true average echo duration. (b) This sample data is used as a pilot study, and now the investigator would like to design a new study to construct a 99% confidence interval with width 0.1. What is the necessary sample size?

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Answer from: lailabirdiemae

a) $0.81-2.62\frac{0.34}{\sqrt{110}}=0.725$

$0.81+2.62\frac{0.34}{\sqrt{110}}=0.895$

So on this case the 99% confidence interval would be given by (0.725;0.895)

b) $n=(\frac{2.58(0.34)}{0.1})^2 =76.95 \approx 77$

So the answer for this case would be n=77 rounded up to the nearest integer

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

$\bar X=0.81$ represent the sample mean

$\mu$ population mean (variable of interest)

s=0.34 represent the sample standard deviation

n=110 represent the sample size

Part a

The confidence interval for the mean is given by the following formula:

$\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}$ (1)

In order to calculate the critical value $t_{\alpha/2}$ we need to find first the degrees of freedom, given by:

df=n-1=110-1=109

Since the Confidence is 0.99 or 99%, the value of $\alpha=0.01$ and $\alpha/2 =0.005$ , and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.005,119)".And we see that $t_{\alpha/2}=2.62$

Now we have everything in order to replace into formula (1):

$0.81-2.62\frac{0.34}{\sqrt{110}}=0.725$

$0.81+2.62\frac{0.34}{\sqrt{110}}=0.895$

So on this case the 99% confidence interval would be given by (0.725;0.895)

Part b

The margin of error is given by this formula:

$ME=z_{\alpha/2}\frac{\sigmas}{\sqrt{n}}$ (a)

And on this case we have that ME =0.1 and we are interested in order to find the value of n, if we solve n from equation (a) we got:

$n=(\frac{z_{\alpha/2} \sigma}{ME})^2$ (b)

We can assume the following estimator for the population deviation $\hat \sigma =s =0.34$

The critical value for 99% of confidence interval now can be founded using the normal distribution. And in excel we can use this formla to find it:"=-NORM.INV(0.005;0;1)", and we got $z_{\alpha/2}=2.58/tex], replacing into formula (b) we got:
[tex]n=(\frac{2.58(0.34)}{0.1})^2 =76.95 \approx 77$

So the answer for this case would be n=77 rounded up to the nearest integer

Answer from: Quest

answer: option a.

step-by-step explanation:

you know that the quadrilateral opqr ( a four sided figure) is inscribed in circle. this means that all its vertices are on the circle.

then, by definition, the opposite angles of the quadrilateral are supplementary, this means that the sum of their measures is 180 degrees.

you can observe in the figure that the angles ∠opq and the angle ∠orq are opposite, therefore:

$m\angle opq+m\angle orq=180\°\\\\m\angle opq+ (2x + 16)\°=180\°$

then, $m\angle opq+ (2x + 16)\°=180\°$ can be used to d to calculate the measure of ∠opq

Quadrilateral opqr is inscribed in circle n, as shown below. which of the following could be used to

Answer from: Quest

false

step-by-step explanation:

false. two legs of the trapezoid would be congruent, but the other two would not. were all the legs congruent, then we'd have a square; were the 2 shorter sides congruent and the 2 longer sides also congruent, we'd have a rectangle.